Length comparison decided in linear time by multi-tape Turing machine

There exists a multi-tape Turing machine $M$ with tape count $k$ and alphabet size $G$ that decides whether the certificate $w$ has length at most the instance $x$:

within a linear number of steps $c_0 (|x| + |w| + 1)$ for some constant $c_0 \in \mathbb{N}$.

The machine operates by simultaneously advancing the heads on tape 0 (holding $x$) and tape 1 (holding $w$) one cell per step until encountering a blank symbol. If the certificate head reaches a blank while the instance head is still reading input bits (or simultaneously reaches a blank), $|w| \le |x|$ holds and the machine writes $\mathbf{1}$ (true) to cell 1 of the verdict tape. If the instance head encounters a blank first, $|w| > |x|$ and the machine writes $\mathbf{0}$ (false). The total step count is bounded by $|x| + |w| + 3$.